Bending of a circular plate with stiff edge lying on a soft functionallygraded elastic layer under the action of arbitrary axisymmetric loading
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1 Beg of ccul ple h ff ege lg o of fucollge elc le ue he co of b ec log Sege S. Volov lee. veo Sege M. zovch 3 Yu-he Wg 4 e S. Vlev Reech Euco ee "Mel o Se Techcl Uve Roov-o-o 344 Ru Iue of Mhec Mechc opue Scece Souhe Feel Uve Roov-o-o 3449 Ru 4 epe of vl Egeeg ol heg Kug Uve T 7 T 3 zovch@gl.co BSTRT ccul ple h co hce fe u ff ege lg o elc hlf-pce coee. The hlf-pce co of of fucoll ge le cog h b vg elc popee hoogeeou elc ube. The ple be ue he co of b ec bue lo epoe fo he elc hlf-pce. e-lcl oluo fo he poble effecve hole ge of geoec elve le hce echcl elc popee of cog ube ffe of he ple popee couce ug couple poc eho []. ppoe lcl epeo fo he coc ee efleco of he ple e pove. uecl eul hog he ulve epeece of he oluo fo he l pee of he poble e obe h hgh peco.. ITROUTIO The poble of ple beg o oopc hoogeeou elc fouo coee []. The oluo couce b povg he coc ee he fo of poe ee h ubeue eeo of he coeffce of epo fo fe lgebc euo. The l poble lo olve ug colloco eho [3] ppoch be o ppoo of he eel b ohogol polol [4] [5]. The covegece of he oluo o he ec oe o 5 Ph.. Seo Reeche 4 Ph.. oce Pofeo 3.Sc. Ph.. He of boo
2 vege. Sgul egul poc eho ee ue o couc oluo effecve fo bg o ll vlue of he geoec pee of he poble [4] [6]. Mo of he o oluo e pplcble ol fo g ple. ve fe pcull hoe ecbe [4] [5] [6] e pplcble ehe fo fleble o g ple. Thee ube of ece vego h e [7] [8]. I h o e ecbe ppoch be o he couple poc eho [] of olvg ce pe of ul egl euo hch llo oe o couc lcl oluo of he poble ufe fo pplcble fo vlue of geoec echcl popee.. MTHEMTI FORMUTIO OF THE PROBEM cul ple of u R co hce h lg o he bou of elc hlf-pce cog of hoogeeou le cog h hce H hoogeeou hlf-pce ube. We ue clcl cooe e φ z hee z pepecul o he ufce of he cog pe hough he cee of he ple. Ple be ue he co of ec bue lo p epoe fo he le. Youg oulu E Poo o ν of he fouo v h eph ccog o he follog E z z H z E z z E co E z H hee E z ν z e b cououl ffeeble fuco. Heefe ee coepo o he cog o he ube epecvel. The le he ube e ue o be glue hou lg: z z u z H : z z u Oue of he puch he ufce co-fee: z R z : z R 3 The ee he plcee vh z. The ue of p ee e he coc ee ue he ple he efleco of he ple l gel oue M M φ. z z We coe o pe of he bou coo o he ege of he ple: Ple h ff ege: R Such bou coo e he clculo of boo u ell ec. hvg ue ff coou. b Fee ege: R 4 ple 5 R R Bou coo 5 e oelg fouo fee-lg ple ec.
3 We ue he follog oo: he plce opeo. 3. SOUTIO OF THE PROBEM 3. ul egl euo ple he Poo o of he ple To euce he poble o he oluo of he ul egl euo e ue he clcl ppoch be o he Hel egl foo echue: The 3 le o he follog: R z W z 6 W 7 R 8 e u ouce he follog oo: W z W z Ug 9 e ee euo 8 he fo: W Subug 7 o e ge: R E R W R e u ouce he eole vble fuco: H u H R R R W u / H u z z H R p p ' R u R hee clcl ffe of he ple pee eole beg ffe of he ple. Fuco u he eel fo of he egl euo epee of he pple log p chceze he coplce of he elc fouo. The eel fo u eul o h ppeg he coc poble of he eo of g p [9]. Og he pe e ge he Fehol egl euo of he f g ove he fuco ρ 3 R Euo euvle o he follog ul egl euo: 3 3 9
4 3 ccog o he Kchhoff ple oel he efleco of he ple h o f he ffeel euo of beg of he ple: p 4 3. Soluo of he poble I he pee ppe e couc he oluo ol fo ce of ple h ff ege ee 4. We epee he efleco fuco e of ee h epec o egefuco of ocllo of ccul ple h fee ege l o Tel []: 5 hee co e he oo of he euo... /. ue o he le of he poble he coc ee Hel fo α c be epeee : 6 Subug 5 6 o 3 e ge he ul egl euo ove he fuco : 7 e u ppl o he f euo 7 he egl opeo U hle o he eco euo 7 e ll ppl he egl opeo U. he eul of h e ob
5 co co g 8 We ue he follog oo: co U g 9 I geel he eel fo α epe o he popee of he ohoogeeou el. I [] ho h α poee he follog popee: 3 3 O E O B hee B e ce co hch vlue epe o he el popee. I ho [] [3] h he eel fo α h hgh peco c be ppoe b he epeo: b P P P P hee b e ce co obe b ppoo he eel fo α. ele ecpo of he poce of eeg coeffce.. b ecbe [3]. I lo ho [] h he oluo of ppoe ul egl euo eulg fo eplcg fo α 8 b ppoo α pocll ec fo boh h hc cog.e. fo λ λ. e u ouce he fuco: co he ug opeol l [4] euo e epee f euo 8 opeo fo: [] g P P 3 The oluo of he ffeel euo 3 fo h he fo: co 4 hee coeffce e b co h ch z z z. Ug epeo 4 euo 8 c be ee follo
6 co co co 5 Iveg euo 5 ug he Foue fo e ob: co co co 6 Iveg he Hel fo foul 6 ug Pevl' e gve: 7 hee: ch h h ch To eee co e ubue he epeo 7 o euo 8. The e of co.....; eee fo he e of le lgebc euo belo hle. b b b b 8 hee ch h
7 oc ee peue pple o he ple p c be epeee follog ee: 9 p p p p 3 Subug o 4 e ge he fe e of le lgebc euo ove : ; 4 4 p 3 hee: ch co h co I pcul: h 4 h Ug euco eho o he fe e 3 e ge he fe e: K p K 4 3 fe eeg pee K fo fe vlue K ubug he o he epeo 6 e fll ge he coc ee fo he epeo 5 e ge he efleco of he ple. Rl oue M gel oue M of he ple c be epeee he follog epeo ove he efleco: R M R M ple ple 33 hee he clcl ffe of he ple.
8 Ug he epeo 5 33 e ge he epeo fo he oue: ple ple ple R M R M K K 34 Fo ce of he ple h fee ege efg b 5 he oluo of he poble peee [5] bu he epeo co p epeo.5.6. Hee e pee he coece epeo: 35 I 36 The of eeg he pee e l o h e ue he pee ppe: e ubue he epeo o he ul egl euo 8 ge he e of le lgebc euo. Vlue of pee coepog o he ple h fee ege e peee []. 4. UMERI RESUTS 4. Sof hoogeeou le e u coe he beg of he ple lg o le h co elc oul E =co ν =co ue he co of ufol bue u lo: p= <. We ue h he le uch ofe h he ube H E E ue pee H E E o chceze ofe of he le. e u coe ce of cog uch ofe h he ube h β=. The uecl eul pove fo ce of he ple h ff ege ee 4. ccog o Gobuov-Poov [] he ple ue o be fleble o of fe u f > ff f <. uecl eple belo e pove fo hee vlue of pee : = =. =5. Fgue co gph of he coc peue ue he ple of eee flebl =5 lg o cog of lge o eee hce λ. I ee h he peue o hc cog fo ce λ= e l ue he cee of he ple = ooocll cee he ppochg he ege of he ple. Fo eee hce of he cog.<λ< he peue ue he cee cee hle he u vlue eche e he ege of he ple = The peue buo fo ll vlue of elve le hce λ h coplce ooooc behvo ee Fgue : e he e po of he coc ego oe c obeve cee ecee peue.e. peue pe. Wh ecee he elve hce of he cog λ hee peue
9 becoe hghe occup le pce ove o he e po of he coc. Relve efleco el of he ple coepog o he peue pove o fgue e llue o he fgue 3. Fgue. Gph of peue buo veu. The gph e peee fo of cog h β = ple of eee flebl =5 oe lge eee cog hcee λ. Fgue. Gph of peue buo veu. The gph e peee fo of cog h β = ple of eee flebl = 5 oe ll cog hcee λ.
10 Fgue 3. Gph of elve efleco of he ple el veu. The gph e peee fo of cog h β = ple of eee flebl = 5. Fgue 4. Gph of peue buo veu. The gph e peee fo of cog h β = fleble ple h = oe ll cog hcee λ. Fo fleble ple he peue ue he cee cee hle e he ege vlue uffcel ecee ee fgue 4 fo ff ple he oppoe uo obeve ee fgue 5. To llue ho he le hce fluece he peue e pove epeece of he vlue.5 fo he elve le hce λ ee fgue 6. The gph e pove fo β= 5 =5. The peue h locl u fo λ=...5. Fo λ>4 he peue lo oe epe o he vlue of
11 β. Fo λ<.5 he peue pccll oe o chge h eceg λ he cog o h h pccll h o effec o he peue ebuo. The peue fo he ple h fee ff ege ee 5 4 e llue fgue 7. I ee h he peue fo he ple h fee boue e gee he ebouhoo of = lle e ege h fo he ple h ff ege. The ulve ffeece ee o foue. Fgue 5. Gph of peue buo veu. The gph e peee fo of cog h β = ff ple h =. oe ll cog hcee λ. Fgue 6. Gph of peue.5 veu λ. The gph e peee fo of cog h β = 5 ple of eee flebl = 5.
12 e he Youg oulu of he cog ve h eph ccog o oe of he follog ule: z z cog : E z co cog : E z co 7 7 H H The peue ue he cee of he ple veu λ peue buo veu fo fucoll-ge cog hoogeeou cog fo β= =5 e peee he fgue 8 9. I ee h he fucoll ge popee of he cog uffcel ebue he peue epecll fo λ=... Fgue 7. Gph of peue buo veu. The gph e peee fo of cog h β = ple of eee flebl = 5 oe ll cog hcee λ. Fgue 8. Gph of peue veu λ. The gph e peee fo of hoogeeou fucoll ge cog h β = ple of eee flebl = 5.
13 Fgue 9. Gph of peue buo veu. The gph e peee fo of cog hoogeeou fucoll ge cog h β = ple of eee flebl = 5 oe lge eee cog hcee λ. 3. OUSIOS lcl epeo fo he coc ee ppeg ue he ple he efleco fuco e couce ug couple poc eho. The eho llo o coe he elc le lg o uch ffe ube. Ug ppoo fo he eel fo of hgh ccuc poble o ob oluo of he poble hch pplcble fo ll poble vlue of λ ffe of he ple. Se eho uccefull pple o e cl of coc poble fo el h fucoll-ge cog [9] [] [6]. The uho colege he uppo of he Ru Fouo fo Bc Reech g o а ol ve S_. S.M. zovch lo colege uppo of he M of Euco Scece of Ru he feo of Govee ge. REFEREES [] zovch S.M. 99 poc oluo of cl of couple euo.. ppl. Mh. Mech. Vol [] Gobuov-Poov M.I. 94 lculo of be ple o elc hlfpce Pl. M. Meh Vol Ru.
14 [3] Shh.S. 97 O clculo of ple beg o elc le. Izv. SSSR. Meh. Tve. Tel 7-76 Ru. [4] leov V.M Voovch I.I. Soloov M Effecve oluo of he poble of clcl beg of ple of fe h o elc hlf-pce. Izv. SSSR. Meh. Tve. Tel Ru. [5] Boov S.V. 8 The oluo of he coc poble fo ccul ple.. pp. Mh. Mech. Vol [6] leov V.M. Soloov M poc poble of he clcl beg of ple of fe beh elc hlf-pce. Sove ppl. Mech. Vol [7] Khl M. 4 Thee-eol elc oluo fo beg of fucoll ge ecgul ple. Eu.. Mech. /Sol Vol [8] Slv.R.. Slve R..M. Goçlve P.B. uecl eho fo l of ple o eole elc fouo. I.. Sol. Suc. Vol [9] zovch S.M. leov V.M. 984 ec poble of eo of ccul e o elc hlf-pce h oufo h epec o eph. Mechc of Sol Vol [] Tel.I. 969 Beg of ccul ple lg o lel efoble fouo. Izv. SSSR. Meh. Tve. Tel 99- Ru. [] zovch S.M. leov V.M. 98 Popee of coplce fuco fo lee cououl oufo hlf-pce Sove Ph. ol. Vol [] Vlev.S. Volov S.S. zovch S.M. eg Y.R. 4 ec coc poble of he heo of elc fo hoogeeou le. ZMM Z. g. Mh. Mech. Vol [3] zovch S.M. Vlev.S. 3 blel poc eho of olvg he egl euo of he coc poble of he oo of elc hlfpce hoogeeou eph. ppl. Mh. Mech. Vol [4] leov V.M. 973 O he oluo of oe cl of ul euo Sove Ph. ol. Vol. 8. [5] zovch S. Vlev. Sevoov I. Tubch I. Evch. blov E. lcl Soluo fo he Beg of Ple o Fucoll Ge e of ople Sucue H. lebch V.. Eeeev e Shell-le Sucue: o-clcl Theoe pplco Spge- Velg Heelbeg 5-8. [6] Vlev. Sevoov I. zovch S. eg Y.R. Too of puch che o veel-oopc hlf-pce h fucoll ge cog. I.. Eg. Sc. Vol
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